Fourth Painlevé Equation and $PT$-Symmetric Hamiltonians

Abstract

This paper is an addendum to earlier papers \cite{R1,R2} in which it was shown that the unstable separatrix solutions for Painlevé I and II are determined by $PT$-symmetric Hamiltonians. In this paper unstable separatrix solutions of the fourth Painlevé transcendent are studied numerically and analytically. For a fixed initial value, say $y(0)=1$, a discrete set of initial slopes $y^{\prime }(0)=b_n$ give rise to separatrix solutions. Similarly, for a fixed initial slope, say $y^{\prime }(0)=0$, a discrete set of initial values $y(0)=c_n$ give rise to separatrix solutions. For Painlevé IV the large-$n$ asymptotic behavior of $b_n$ is $b_n\sim B_{\mathrm{IV}}{n}^{3/4}$ and that of $c_n$ is $c_n\sim C_{\mathrm{IV}}{n}^{1/2}$. The constants $B_{\mathrm{IV}}$ and $C_{\mathrm{IV}}$ are determined both numerically and analytically. The analytical values of these constants are found by reducing the nonlinear Painlevé IV equation to the linear eigenvalue equation for the sextic $PT$-symmetric Hamiltonian $H=\frac{1}{2}{p}^2+\frac{1}{8}{x}^6$.

9 pages, 9 figures. arXiv admin note: substantial text overlap with arXiv:1502.04089, added references

References in corpus (5)

• Complex oscillator and Painlevé IV equation
• Nonlinear eigenvalue problems for generalized Painlevé equations
• Nonlinear eigenvalue problems
• On "Nonlinear eigenvalue problems"
• A Dynamical Systems Approach to The Fourth Painleve Equation

Cited by in corpus (1)

• The Tracy-Widom distribution at large Dyson index